Generally, two dislocations with the same sign on the same slip plane would repel each other, whereas dislocations with the opposite signs would attract each other.

Let us assume that two parallel edge dislocations of the same sign (both positive edge in the case) are on the same slip plane, as illustrated in Figure 4.16a. If the dislocations come very close together, the configuration can be assumed to be a dislocation with double the Burgers vector (i. e., 2b) of the individual dislocations, and the elastic strain energy would be given by aG(2b)2 or 4aGb2. However, when they are separated by a large distance, they have a total energy of 2aGb2. The dislo­cation configuration with a smaller Burgers vector is more stable. Thus, the disloca­tions would repel each other. However, when two edge dislocations of the opposite signs lie on the same slip plane (one positive and another negative) as illustrated in Figure 4.16b, the effective Burgers vector is zero, and thus the elastic strain energy becomes zero. As in this way the dislocation configuration reduces their energy, the two dislocations will be attracted to each other and will get annihilated. Similarly, while the like screw dislocations on the slip plane will repel each other, the unlike dislocations will attract each other.

Now let us consider the case of the interaction between two parallel screw dislo­cations (not necessarily on the same slip plane). As the screw dislocation has a radi­ally symmetrical stress field, the force between the two dislocations is given by the following equation and depends only on the distance of separation.

Gb2

Fi = tgzb = -—. (4.18)

2pr

This interaction force is repulsive between two like screw dislocations and attractive between two unlike dislocations. This can easily be shown by evaluating the force on a screw dislocation using Peach-Koehler formula (Eq. (4.17b)) due to the stress fields from the second dislocation using Eq. (4.10a) (or using forces in Cartesian coordinates).

The interaction between two edge dislocations is much more complex because of their more complicated stress fields. If we consider one of the edge dislocations to be lying parallel to the z-direction with Burgers vector parallel to the x-direction, the interaction would change from attractive to repulsive and vice versa depending on how the two dislocations are positioned with respect to each other. The various cases are shown in Figure 4.17 in terms of coordinates.

The arrows in Figure 4.17 correspond to forces in the x-direction, while the forces along the y-direction are repulsive between two like edge dislocations although they cannot glide in the y-direction. The edge dislocations can move along the y-direction only by nonconservative climb that involves vacancies and/or atoms move away/to the dislocation core. From Figure 4.17, certain simple cases can be derived. Figure 4.18a shows an edge dislocation parallel to another like edge dislocation just vertically above its slip plane. This is a stable configuration. This type of configuration is found in the low angle tilt boundaries (see Figure 2.37). Two stable forms of dipole pairs are shown in Figure 4.18b where two edge dislocations of opposite signs glide past each other in parallel slip planes at low applied stresses.

& Example 4.6

Derive the force between two like dislocations lying along the z-axis. Solution

?1 : b1 — b1fe, t1 — k and?2 : b2 — b2k, t2 — k.

Force on dislocation 2 due to dislocation 1 : F2 — (a(1) ■ b2)xt2.

0 0 aXz

Recall stresses around screw dislocations : a(1) =

4.2.6